65842961-Open Source MATLAB Implementation of COnsistent Discretisations on Complex Grids

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Open Source MATLAB Implementation of
Consistent Discretisations on Complex Grids
Knut–Andreas Lie Stein Krogstad Ingeborg S. Ligaarden
Jostein R. Natvig Halvor Møll Nilsen B˚ard Skaflestad
Received: date / Accepted: date
Abstract
Accurate geological modelling of features such as faults, fractures or
erosion requires grids that are flexible with respect to geometry. Such grids
generally contain polyhedral cells and complex grid cell connectivities.
The grid representation for polyhedral grids in turn affects the efficient
implementation of numerical methods for subsurface flow simulations. It
is well known that conventional two-point flux-approximation methods
are only consistent for K orthogonal grids and will therefore not converge
in the general case. In recent years, there has been significant research
into consistent and convergent methods, including mixed, multipoint, and
mimetic discretisation methods. Likewise, so-called multiscale methods
based upon hierarchically coarsened grids have received a lot of attention.
The paper does not propose novel mathematical methods but instead
presents an open-source Matlab® toolkit that can be used as an efficient
test platform for (new) discretisation and solution methods in reservoir
simulation. The aim of the toolkit is to support reproducible research
and simplify the development, verification and validation, and testing
and comparison of new discretisation and solution methods on general
unstructured grids, including in particular corner-point and 2.5D PEBI
grids. The toolkit consists of a set of data structures and routines for
creating, manipulating, and visualising petrophysical data, fluid models,
and (unstructured) grids, including support for industry-standard input
formats, as well as routines for computing single and multiphase (incom-
pressible) flow. We review key features of the toolkit and discuss a generic
mimetic formulation that includes many known discretisation methods,
including both the standard two-point method as well as consistent and
convergent multipoint and mimetic methods. Apart from the core routines
and data structures, the toolkit contains add-on modules that implement
more advanced solvers and functionality. Herein, we show examples of
multiscale methods and adjoint methods for use in optimisation of rates
and placement of wells.
1
1 Introduction
Reliable computer modelling of subsurface flow is much needed to overcome
important challenges such as sustainable use and management of the earth’s
groundwater systems, geological storage of CO2 to mitigate the anthropological
increases in the carbon content of the atmosphere, and optimal utilisation of
hydrocarbon reservoirs. Indeed, the need for tools that help us understand
flow processes in the subsurface is probably greater than ever, and increasing.
More than fifty years of prior research in this area has led to some degree of
agreement in terms of how subsurface flow processes can be modelled adequately
with numerical simulation technology.
To describe the subsurface flow processes mathematically, two types of mod-
els are needed. First, one needs a mathematical model that describes how fluids
flow in a porous medium. These models are typically given as a set of partial
differential equations describing the mass-conservation of fluid phases, accom-
panied by a suitable set of constitutive relations. Second, one needs a geological
model that describes the given porous rock formation (the reservoir). The geo-
logical model is realised as a grid populated with petrophysical or hydrological
properties that are used as input to the flow model, and together they make
up the reservoir simulation model. The geological model must also describe the
geometry of the reservoir rock and in particular model geological horizons and
major faults. This requires grids that are flexible with respect to geometry (and
topology). Stratigraphic grids have been popular for many years and are the
current industry standard. These grids are formed by extruding areal grids de-
fined along geological surfaces to form volumetric descriptions. However, more
complex methods based on unstructured grids are gaining in popularity as a
means to modelling complex fault systems, horizontal and multilateral wells,
etc. In either case, grids representing realistic reservoirs generally contain poly-
hedral cells and complex grid cell connectivities. The grid representation for
polyhedral grids in turn affects the efficient implementation of numerical meth-
ods for subsurface flow simulations.
The industry-standard for discretising flow equations is the two-point flux-
approximation method, which for a 2D Cartesian grid corresponds to a standard
five-point scheme for the elliptic Poisson equation. Although widely used, this
method is convergent only if each grid cell K-orthogonal. For hexahedral grids,
this means that each cell is a parallelepiped and ~nijKi~nik = 0 in all grid cells
i (here, K is the permeability tensor in cell i and ~nij and ~nik denote normal
vectors into two neighbouring cells). Ensuring K-orthogonality is difficult when
representing particular geological features like sloping faults, horizontal wells,
etc. Hence, there has in recent years been significant research into mixed [8],
multipoint [5], and mimetic [9] discretisation methods that are all consistent and
convergent on rougher grids. Herein, we will focus on low-order, cell-centred
methods that do not require specific reference elements and thus can be applied
to grids with general polygonal and polyhedral cells.
Another major research challenge is the gap between simulation capabilities
and the level of detail available in current geological models. Despite an as-
2
tonishing increase in computer power, and intensive research on computation
techniques, commercial reservoir simulators can seldom run simulations directly
on highly resolved geological grid models that may contain from one to a hun-
dred million cells. Instead, coarse-grid models with grid-blocks that are typically
ten to a thousand times larger are built using some kind of upscaling of the geo-
physical parameters [16, 13]. How one should perform this upscaling is not
trivial. In fact, upscaling has been, and probably still is, one of the most active
research areas in the oil industry. Lately, however, so-called multiscale methods
[17, 15, 18] have received a lot of attention. In these methods, coarsening and
upscaling needed to reduce the number of degrees of freedom to a level that
is sufficient to resolve flow physics and satisfy requirements on computational
costs is done implicitly by the simulation method.
A major goal of the activities in our research group is to develop efficient
simulation methodologies based on accurate and robust discretisation methods;
in particular, we have focused on developing multiscale methods. To this end,
we need a toolbox for rapid prototyping of new ideas that enables us to eas-
ily test the new implementations on a wide range of models, from small and
highly idealised grid models to large models with industry-standard complexity.
When developing new computational methodologies, flexibility and low devel-
opment time is more important than high code efficiency, which will typically
only be fully achieved after the experimental programming is completed and
ideas have been thoroughly tested. For a number of years, we have therefore
divided our code development in two parts: For prototyping and testing of new
ideas, we have used Matlab, whereas solvers aimed at high computational per-
formance have been developed in a compiled language (i.e., using FORTRAN,
C, or generic programming in C++).
This has resulted in a comprehensive set of routines and data structures for
reading, representing, processing, and visualising unstructured grids, with par-
ticular emphasis on the corner-point formatused within the petroleum industry
and hierarchical grids used in multiscale methods. To enable other researchers
to benefit from our efforts, these routines have been gathered in the Matlab
Reservoir Simulation Toolbox (MRST), which is released under the GNU Gen-
eral Public License (GPL). The first releases are geared towards single- and
two-phase flow and contain a set of mimetic and multiscale flow solvers and a
few simple transport solvers capable of handling general unstructured, polyhe-
dral grids.
The main purpose of this paper is to present MRST and demonstrate its
flexibility and efficiency with respect to different grid formats, and in particular
hierarchical grids used in multiscale methods. Secondly, we present a class of
mimetic methods that incorporates several well-known discretisation methods
as special cases on simple grids while at the same time providing consistent dis-
cretisation on grids that are not K-orthogonal. Finally, we discuss how generic
implementations of various popular methods for pressure and transport ease
the study and development of advanced techniques such as multiscale methods,
flow-based gridding, and applications such as optimal control or well placement.
3
2 The MATLAB Reservoir Simulation Toolbox
The toolbox has the following functionality for rapid prototyping of solvers for
flow and transport:
Grids: a common data structure and interface for all types of grids (unstruc-
tured representation); no grid generator but grid factory routines for recti-
linear grids, triangular and tetrahedral grids, 2D Voronoi grids, extrusion
of areal grids to 2.5D volumetric grids, etc; tutorial examples and a few
realistic data sets
Input and output: routines for reading and processing industry-standard in-
put files for grids, petrophysical parameters, fluid models, wells, boundary
conditions, simulation setup, etc.
Parameters: a data structure for petrophysical parameters (and a few, very
simplified geostatistical routines); common interface to fluid models (Ver-
sion 2011a supports incompressible fluid models, but in-house develop-
ment version also has support for compressible black-oil fluids, which will
likely be released in future versions of MRST); routines for setting and
manipulating boundary conditions, sources/sinks, well models, etc
Units: MRST works in strict SI units but supports conversion to/from other
unit systems like field-units, etc. Unless reading from an industry-standard
input format, the user is responsible for explicit conversion and consistency
of units1.
Reservoir state: data structure for pressure, fluxes, saturations, . . .
Postprocessing: visualisation routines for scalar cell and face data, etc
Solvers: the toolbox contains several flow and transport solvers which may be
readily combined using an operator splitting framework. In particular, we
provide an implementation of the multiscale mixed finite-element method
[4], working on unstructured, polyhedral grids
Linear algebra: MRST relies on Matlab’s builtin linear solvers but these can
easily be replaced by specialised solvers using standard Matlab conven-
tions for doing so.
We will now go into more details about some of the components outlined
above. All code excerpts and explicit statements of syntax refer to release 2011a
of MRST. Complete scripts and the data necessary to run most of the examples
in the paper can be downloaded from the MRST webpage [26]. The interested
reader should also review the tutorials included in the current release.
1All examples considered herein are computed using SI units, but other units may be used
when reporting parameters and solutions.
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cells.faces = faces.nodes = faces.neighbors =
1 10 1 1 0 2
1 8 1 2 2 3
1 7 2 1 3 5
2 1 2 3 5 0
2 2 3 1 6 2
2 5 3 4 0 6
3 3 4 1 1 3
3 7 4 7 4 1
3 2 5 2 6 4
4 8 5 3 1 8
4 12 6 2 8 5
4 9 6 6 4 7
5 3 7 3 7 8
5 4 7 4 0 7
5 11 8 3
6 9 8 5
6 6 9 3
6 5 9 6
7 13 10 4
7 14 10 5
: : : :
Figure 1: Illustration of the cell and faces fields of the grid structure: cell
numbers are marked by circles, node numbers by squares, and face numbers
have no marker. A zero value in face.neighbors means that the faces lies at an
outer boundary and hence only has one neighbour.
2.1 Grids
A key requirement for MRST is that it should support a large variety of grid
types. To avoid having a large number of different and potentially incompatible
grid representations, all grids in MRST are assumed to consist of a set of non-
overlapping polyhedral cells, where each cell can have a varying number of planar
faces that match the faces of the cell’s neighbours. Grids with non-matching
faces, e.g., corner-point and other extruded grids, are therefore converted into
matching grids by splitting non-matching faces into a set of matching (sub)faces.
All grids are stored in a general format in which we explicitly represent cells,
faces, and vertices and the connections between cells and faces. Hence, we have
sacrificed some of the efficiency attainable by exploiting special structures in a
particular grid type for the sake of generality and flexibility.
The grid structure in MRST contains three fields—cells, faces, and nodes—
that specify individual properties for each individual cell/face/vertex in the grid.
The nodes structure is simple, it contains the number of nodes Nn and an Nn×d
array of physical nodal coordinates in Rd. The cells structure contains the
number of cells Nc, an array cells.faces giving the global faces connected to
a given cell, and an indirection map into cells.faces that gives the number of
faces per cell. The cells.faces array has nf × 2 elements defined so that if
cells.faces(i,1)==j, then global face cells.faces(i,2) is connected to global
cell number j. To conserve memory, only the last column is stored, whereas
the first column can be derived from a run-length encoding of the indirection
map. The cells structure may optionally contain an array cells.indexMap
that maps internal cell indexes to external cell indexes, which is useful e.g.,
if the model contains inactive cells2. Likewise, the faces structure contains the
2Inactive cells are specified in industry-standard input format using special keywords (e.g.,
ACTNUM in the widely used ECLIPSE standard). Alternatively, inactive cells in clist can
5
dx = 1−0.5*cos((−1:0.1:1)*pi);
x = −1.15+0.1*cumsum(dx);
G = tensorGrid (x , sqrt (0:0 .05 :1));
plotGrid (G );
load seamount % Matlab standard dataset
g = pebi( triangleGrid ([ x (:) y (:)]));
G = makeLayeredGrid (g , 5);
plotGrid (G ), view(−40, 60),
G = processGRDECL ( ...
simpleGrdecl ([20, 10, 5], 0.12 ));
plotGrid (G , 'FaceAlpha',0.8);
plotFaces (G , find ( G.faces.tag >0), ...
'FaceColor','red' );
view (40,40), axis off
grdecl = readGRDECL ( 'SAIGUP.GRDECL');
G = processGRDECL ( grdecl );
plotGrid (G , 'EdgeAlpha',0.1);
view(−80,50), axis tight off
Figure 2: Examples of grids and the MRST statements necessary to create them.
The upper-left plot shows a standard tensor-product Cartesian grid. In the upper-
right plot we 2.5D PEBI grid extruded from the unstructured point set in the data file
seamount.mat that is distributed with Matlab. The lower-left plot shows an example
of a corner-point grid with a single sloping fault and wavy top- and bottom surfaces.
The lower-right plot shows one of the grid models supplied together with MRST.
number of global faces, an array faces.nodes of vertices connected to each face,
an indirection map, and an array neighbors giving neighbouring information
(face i is shared by global cells neighbors(i,1) and neighbors(i,2)). In addition,
the grid may contain a label type which records the grid constructionhistory.
Finally, grids supporting an underlying logical Cartesian structure also include
the field cartDims. The grid structure is illustrated in Figure 1 for a triangular
grid with eight cells.
MRST contains several grid-factory routines for creating structured grids,
including regular Cartesian, rectilinear, and curvilinear grids, as well as unstruc-
tured grids, including Delaunay triangulations and Voronoi grids, and 3D grids
be explicitly removed through a call to removeCells(G,clist).
6
created by extrusion of 2D shapes. Most important, however, is the support
for the industry-standard corner-point grids given by the ECLIPSE input deck.
In Figure 2 we show four examples of grids and the commands necessary to
create and display them. The rectilinear grid is generated by the grid-factory
routine tensorGrid, which takes one vector of grid nodes per spatial dimension
as input and creates the corresponding tensor-product mesh. To generate the
2.5D PEBI grid, we start with an unstructured point set (seamount.mat) and
use triangleGrid to generate a Delauny grid, from which a 2D Voronoi grid is
generated using pebi. Then this areal grid is extruded to a 3D model consisting
of five layers using the routine makeLayeredGrid. In the third plot, we have used
the routine simpleGrdecl to generate an example of an ECLIPSE input stream
and then processGRDECL to process the input stream and generate a corner-point
grid. The lower-right plot shows a realistic reservoir model where we have used
readGRDECL to read the grid section of an ECLIPSE input deck; the routine reads
the file and creates an input stream on the same format as simpleGrdecl.
As we will see below, specific discretisation schemes may require other prop-
erties not supported by our basic grid class: cell volumes, cell centroids, face
areas, face normals, and face centroids. Although these properties can be com-
puted from the geometry (and topology) on the fly, it is often useful to pre-
compute and include them explicitly in the grid structure G. This is done by
calling the generic routine G=computeGeometry(G).
2.2 Petrophysical Parameters
All flow and transport solvers in MRST assume that the rock parameters are rep-
resented as fields in a structure. Our naming convention is that this structure is
called rock, but this is not a requirement. The fields for porosity and permeabil-
ity, however, must be called poro and perm, respectively. Whereas petrophysical
parameters are often supplied for all cells (active and inactive) in a model, the
rock and grid structures in MRST only represent the active cells. The porosity
field rock.poro is therefore a vector with one value for each active cell in the cor-
responding grid model. For models with inactive cells, the field cells.indexMap
in the grid structure contains the indices of the active cells sorted in ascending
order. If p contains porosity values for all cells, this porosity distribution is
assigned to the rock structure by the call rock.poro = p(G.cells.indexMap).
The permeability field rock.perm can either contain a single column for an
isotropic permeability, two or three columns for a diagonal permeability (in
two and three spatial dimensions, respectively), or three or six columns for
a symmetric, full tensor permeability (in two and three spatial dimensions,
respectively). In the latter case, cell number i has the permeability tensor
Ki =
[
K1(i) K2(i)
K2(i) K3(i)
]
, Ki =
K1(i) K2(i) K3(i)K2(i) K4(i) K5(i)
K3(i) K5(i) K6(i)
 ,
where Kj(i) is the entry in column j and row i of rock.perm. Full-tensor, non-
symmetric permeabilities are currently not supported in MRST. In addition to
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0 10 20 30 40 50
0
5
10
15
20
 
 
0.2 0.25 0.3 0.35 0.4
 
 
25 50 100 200 400 800
Figure 3: Two examples of MRST’s simplified geostatistics. The left plot shows a
50 × 20 porosity field generated as a Gaussian field with a larger filter size in the x-
direction than in the y-direction. The right plot shows a stratigraphic grid with a single
fault and four geological layers, each with a log-normal permeability distribution.
porosity and permeability, MRST supports a field called ntg that represents the
net-to-gross ratio and consists of either a scalar or a single column with one
value per active cell.
Given the difficulty of measuring rock properties, it is common to use geo-
statistical methods to make realisations of porosity and permeability. MRST
contains two very simplified methods for generating geostatistical realisations.
As a simple approximation to a Gaussian field, we generate a field of indepen-
dent, normally distributed variables and convolve it with a Gaussian kernel.
This method is used in two different routines, gaussianField and logNormLayers
that are illustrated in the following examples.
Example 1 (Random Petrophysical Variables) First, we generate the poros-
ity φ as a Gaussian field taking values in the interval [0.2, 0.4]. To get a
crude approximation to the permeability-porosity relationship, we start with the
Carman–Kozeny relation
K =
1
2τA2v
φ3
(1− φ)2 ,
and assume that our medium is made up of uniform spherical grains of diameter
dp = 10µm, for which the specific surface area is Av = 6/dp. Assuming further
that τ = 0.81 gives us an explicit formula for calculating the isotropic perme-
ability K from the porosity φ. Then, petrophysical parameters can be generated
as follows (the resulting porosity field is shown in the left plot of Figure 3):
G = cartGrid([50 20]);
p = gaussianField(G.cartDims, [0.2 0.4], [11 3], 2.5); p = p(:);
rock.poro = p;
rock.perm = p.ˆ3.*(1e−5)ˆ2./(0.81*72*(1−p).ˆ2);
Next, we will use the same Gaussian field methodology to generate layered
realisations, for which the permeability in each geological layer is independent
8
of the other layers and log-normally distributed. Each layer can be represented
by several grid layers in the vertical direction. Here, we generate a stratigraphic
grid with wavy geological faces and a single fault and specify four geological layers
with mean values of 100 mD, 400 mD, 50 mD, and 350 mD from top to bottom
(stratigraphic grids are numbered from the top and downward)
G = processGRDECL(simpleGrdecl([50 30 10], 0.12));
K = logNormLayers(G.cartDims, [100 400 50 350], 'indices', [1 2 5 7 11]);
The layers are represented with one, three, two, and four grid layers, respec-
tively, in the vertical direction. The resulting permeability is shown in the right
plot of Figure 3.
Using smoothed Gaussian fields to generate random petrophysical variables
is, of course, a gross simplification of geostatistics. For more realistic dis-
tributions of petrophysical parameters, the reader should consider using e.g.,
GSLIB [12] or commercial software for geological modelling.
2.3 Discretisation of Flow Equations
To keep technical details at a minimum, we will in the following consider a
simplified set of single-phase flow equations,
∇ · ~v = q, ~v = −K∇p, in Ω ⊂ Rd. (1)
Here, ~v denotes the Darcy velocity, p pressure, and K permeability. All exter-
nal boundaries ∂Ω are equipped with either prescribed pressure (Dirichlet) or
prescribed flux (Neumann) boundary conditions. Let ui be the vector of out-
ward fluxes of the faces of Ωi and let pi denote the pressure at the cell centre
and pii the face pressures. Discretisation methods used for reservoir simulation
are constructed to be locally conservative and exact for linear solutions. Such
schemes can be written in a form that relates these three quantities through a
matrix T i of one-sided transmissibilities,
ui = T i(eipi − pii), ei = (1, . . . , 1)T. (2)
Examples include the two-point flux-approximation method [7], the lowest-order
mixed finite-element methods [8], multipoint flux approximation schemes [6, 14,
5], and recently developed mimetic finite-differencemethods [9]. Two-point
discretisations give diagonal transmissibility matrices and are not convergent
for general grids. Mixed, multipoint, and mimetic methods are consistent and
convergent on non-orthogonal grids, but lead to full matrices T i. Such schemes
will be discussed in more detail in Section 3; for now we only assume that
there exists a consistent scheme of the form (2) that is convergent for fully
unstructured, polyhedral grids.
In the following, we only consider schemes that may be written in hybridised
mixed form, although MRST also supports mixed forms. Note that this re-
striction, which excludes some multipoint schemes, is only imposed to ease the
9
presentation and give a uniform formulation of a large class of schemes. The
underlying principles may be applied to any reasonable scheme. By augmenting
(2) with flux and pressure continuity across cell faces, we obtain the following
linear system [8]  B C DCT 0 0
DT 0 0
 u−p
pi
 =
0q
0
 . (3)
Here, the first row in the block-matrix equation corresponds to Darcy’s law in
the form (2) for all grid cells, the second row corresponds to mass conservation
for all cells, whereas the third row expresses continuity of fluxes for all cell faces.
Thus, u denotes the outward face fluxes ordered cell-wise (fluxes over interior
faces and faults appear twice with opposite signs), p denotes the cell pressures,
and pi the face pressures. The matrices B and C are block diagonal with each
block corresponding to a cell. For the two matrices, the i’th blocks are given as
T−1i and ei, respectively. Similarly, each column of D corresponds to a unique
face and has one (for boundary faces) or two (for interior faces) unit entries
corresponding to the index(s) of the face in the cell-wise ordering.
The hybrid system (3) can be solved using a Schur-complement method and
Matlab’s standard linear solvers or third-party linear system solver software
such as AGMG [29]. A block-wise Gaussian elimination for (3) yields a positive-
definite system (the Schur complement) for the face pressures,(
DTB−1D − F TL−1F )pi = F TL−1q, (4)
where F = CTB−1D and L = CTB−1C. Given the face pressures, the cell
pressures and fluxes can be reconstructed by back-substitution, i.e., solving
Lp = q + Fpi, Bu = Cp−Dpi.
Here, the matrix L is by construction diagonal and computing fluxes is therefore
an inexpensive operation. It is also worth noting that we only need B−1 in the
solution procedure above. Many schemes—including the mimetic method, the
MPFA-O method, and the standard two-point scheme—yield algebraic approx-
imations for the B−1 matrix. Thus, (3) encompasses a family of discretisation
schemes whose properties are determined by the choice of B, which we will
discuss in more detail in Section 3.1.
2.4 Putting it all Together
In this section, we will go through a very simple example to give an overview of
how to set up and use a discretisation as introduced in the previous section to
solve the single-phase pressure equation
∇ · ~v = q, ~v = −K
µ
[∇p+ ρg∇z]. (5)
First, we construct a Cartesian grid of size nx × ny × nz cells, where each cell
has dimension 1 × 1 × 1 m and set an isotropic and homogeneous permeability
of 100 mD, a fluid viscosity of 1 cP, and a fluid density of 1014 kg/m3:
10
nx = 20; ny = 20; nz = 10;
G = computeGeometry(cartGrid([nx, ny, nz]));
rock.perm = repmat(100 * milli*darcy, [G.cells.num, 1]);
fluid = initSingleFluid('mu', 1*centi*poise, 'rho', 1014*kilogram/meterˆ3);
gravity reset on
The simplest way to model inflow or outflow from the reservoir is to use a fluid
source/sink. Here, we specify a source with flux rate of 1 m3/day in each grid
cell.
c = (nx/2*ny+nx/2 : nx*ny : nx*ny*nz) .';
src = addSource([], c, ones(size(c)) ./ day());
Flow solvers in MRST automatically assume no-flow conditions on all outer
(and inner) boundaries; other types of boundary conditions need to be specified
explicitly. To draw fluid out of the domain, we impose a Dirichlet boundary
condition of p = 10 bar at the global left-hand side of the model.
bc = pside([], G, 'LEFT', 10*barsa());
Here, the first argument has been left empty because this is the first bound-
ary condition we prescribe. The left plot in Figure 4 shows the placement of
boundary conditions and sources in the computational domain. Next, we con-
struct the system components for the hybrid mimetic system (3), with a mimetic
discretisation, based on input grid and rock properties.
S = computeMimeticIP(G, rock, 'Verbose', true);
Rather than storing B, we store its inverse B−1. Similarly, the C and D blocks
are not represented in the S structure; they can easily be formed explicitly
whenever needed, or their action can easily be computed.
Finally, we compute the solution to the flow equation. To this end, we must
first create a state object that will be used to hold the solution and then pass
this objects and the parameters to the incompressible flow solver.
rSol = initResSol(G, 0);
rSol = solveIncompFlow(rSol, G, S, fluid,'src', src, 'bc' , bc);
p = convertTo(rSol.pressure(1:G.cells.num), barsa() );
Having computed the solution, we convert the result back to the unit bars. The
right plot in Figure 4 shows the corresponding pressure distribution, where we
clearly can see the effects of boundary conditions, source term, and gravity.
The same basic steps can be repeated on (almost) any type of grid; the only
difference is placing the source terms and how to set the boundary conditions,
which will typically be more complicated on a fully unstructured grid. We will
come back with more examples later in the paper, but then we will not explic-
itly state all details of the corresponding MRST scripts. Before giving more
examples, however, we will introduce the multiscale flow solver implemented in
MRST.
11
0
5
10
15
20
0
5
10
15
20
0
5
10
 
 
10 10.2 10.4 10.6 10.8 11 11.2
Figure 4: Example of a simple flow driven by a column of source cells and a Dirichlet
boundary condition. The left plot shows the model setup and the right plot the
corresponding pressure solution.
2.5 Two-Phase Flow
Two-phase incompressible flow of a wetting and non-wetting fluid can be de-
scribed by the following system of equations [10] (or see the derivation in [1]):
∇ · ~v = q, ~v = −K[λ∇p+ (λwρw + λnρn)g∇z], (6)
φ
∂sw
∂t
+∇ ·
(
fw(sw)
[
~v + λn(ρn − ρw)gK∇z
])
= qw. (7)
Here, ρα denotes the density, λα the mobility, and fα = λα/λ the fractional
flow of phase α, where λ = λn+λw is the total mobility. The industry-standard
approach is to use implicit discretisations and solve (6)–(7) as a fully-coupled
system. In MRST, on the other hand, our goal has been to obtain maximum
flexibility in combining different types of solvers. Hence, the toolbox assumes
a sequential solution strategy: First, (6) is solved with fixed saturation val-
ues to provide pressure and fluxes, and then the fluxes are used to evolve the
saturations according to (7). If necessary, the two steps can be iterated until
convergence in a suitable norm.
All flow solvers in MRST are fully applicable to the two-phase flow equation
(6). MRST supports two basic saturation solvers that both use a single-point
upwind discretisation (i.e., the single-point upstream-mobility scheme for unidi-
rectional flow). Dropping subscripts to denote phases and assuming no gravity,
they can be written in the form
sn+1i = s
n
i +
∆t
φi|ci|
(
max(qi, 0) + f(smi ) min(qi, 0)
)
− ∆t
φi|ci|
(∑
j
[
f(smi ) max(vij , 0) + f(s
m
j ) min(vij , 0)
])
, (8)
12
pi πk
Ak~cik
~nk
Figure 5: The quantities used to define the mimetic inner product or equivalently
the transmissibility in a single polyhedral cell: pi cell pressure, pik face pressure, ~nk
normal of face, Ak area of face and ~cik vectorfrom cell centroid to face centroid.
Here, si is the average saturation in grid cell ci, vij denotes the flux over the face
between cells i and j. For m = n, the scheme is explicit, whereas for m = n+ 1,
we obtain an implicit scheme that is solved by a Newton–Raphson method. For
systems with gravity forces, MRST uses standard upstream mobility weighing;
that is, the upwind direction is determined independently for each phase using
the phase velocities ~vα.
The observant reader may have noticed that the capillary pressure is missing
in our two-phase model. In (6), capillary effects can be included by defining the
pressure as the global pressure, p = pn− pc, where the so-called complementary
pressure [10] is defined through the relation ∇pc = fw∇(pn − pw).
So far, the paper has given a quick overview of the basic functionality and
solvers in MRST Release 2011a. In the next section, we will go into more de-
tails about consistent and convergent discretisations on unstructured polyhedral
grids, before we end the paper with an overview of how the resulting solvers can
be applied to more advanced examples, including solvers and functionality that
are not yet released publicly.
3 Mimetic Discretisation Methods
In this section we will discuss the mimetic method in more detail. We start by
discussing the inner product, which can be used to design the properties of the
method. Then we give a short discussion of Peaceman-type well models for the
method.
The mimetic method (see [9]) is defined in terms of a local inner product M
or equivalently an inverse inner product (or transmissibility matrix) T , which
gives the cell-based discretisation of Darcy’s law
Mu = epi − pi, ~e = (1, . . . , 1)T, u = T (epi − pi). (9)
Here pi is the pressure at the face centroids and p the cell pressure or equivalently
13
the pressure at the cell centroid as illustrated in Figure 5. Mimetic methods are
constructed so that they are exact for linear pressure fields and give a symmetric
positive-definite matrix M . In addition, the methods use discrete pressure and
fluxes associated with cell and face centroids, respectively, and consequently
resemble finite-difference methods.
A linear pressure field can be written in the form p = ~x ·~a+ b for a constant
vector ~a and scalar b, giving a Darcy velocity equal ~v = −K~a. Let ~nk denote the
area-weighted normal vector to face number k and ~cik be the vector pointing
from the cell centroid to the corresponding face centroid, as seen in Figure 5.
Then the flux and pressure drop are given by
uk = −~nkK~a, pi − pij = ~cik · ~a. (10)
Inserting this into (9), we see that the matrices M and T must satisfy the
following consistency conditions
MNK = C, NK = TC, (11)
where each row cTi of the matrix C corresponds to ~cik, and each row n
T
i of N
corresponds to ~nk, see [9] for the discrete flux case.
From the local discretisation on each cell, the global stiffness matrix B in (3)
is assembled as a block-diagonal matrix in which each block is the inner product
of the corresponding cell. For two-phase flow, the inner product is multiplied by
the inverse total mobility for each cell. As we have seen, the system (3) can be
reduced to a linear system that only involves the face pressures pi, using a trans-
formation that requires the computation of B−1, which is a block matrix that
can be assembled from the inverse inner product T . In the following, we describe
a few inner products (and inverse inner products) while emphasising aspects of
implementation and discuss specific properties of the different discretisations.
The implementation can be found in computeMimeticIP.m in MRST [26].
3.1 Inner Products
In the original method [9], inner products of discrete velocities are considered.
In reservoir simulation however, it is more common to consider the face fluxes
as unknowns. Accordingly, we will henceforth consider inner products of fluxes
rather than velocities. We note here that the relation between the two is trivial,
as an inner product of velocities becomes an inner product for fluxes by pre-
and post-multiplying by the inverse area of the corresponding faces. Let A be
the diagonal matrix with aii the face area of the i-th face. Then the flux inner
product Mflux is related to the velocity inner product Mvel through
Mflux = A−1MvelA−1. (12)
Henceforth we will only consider inner products for fluxes.
To yield a consistent discretisation of (1), an inner product matrix M or an
inverse inner product T must result in a discretisation that is exact for linear
14
pressures, i.e., fulfils (11). To derive a family of valid solutions, we first observe
the following key geometrical property (see [9])
CTN = diag(|Ωi|), (13)
which relates C and N as follows on general polyhedral cells. Multiplying the
first equation of (11) by K−1CTN , we derive the relation
MN =
1
|Ωi|CK
−1CTN ,
from which it follows that the family of valid solutions has the form
M =
1
|Ωi|CKC
T +M2, (14)
in which M2 is a matrix defined such that M is symmetric positive definite and
M2N = 0. Any symmetric and positive definite inner product fulfilling these
requirements can be represented in the compact form
M =
1
|Ωi|CK
−1CT +Q⊥NSMQ
⊥
N
T
, (15)
where Q⊥N is an orthonormal basis for the null space of N
T, and SM is any
symmetric, positive-definite matrix. In the code and the following examples,
however, we will instead use a null space projection P⊥N = I −QNQNT, where
QN is a basis for the space spanned by the columns of N . Similarly, the inverse
inner products take the form
T =
1
|Ωi|NKN
T +Q⊥CSQ
⊥
C
T
, (16)
whereQ⊥C is an orthonormal basis for the nullspace ofC
T and P⊥C = I−QCQCT
is the corresponding nullspace projection.
The matrices M and T in (15) and (16) are evidently symmetric so only an
argument for the positive definiteness is in order. Writing M in (15) as M =
M1 +M2, positive semi-definiteness of M1 and M2 imply that M is positive
semi-definite. Let z be an arbitrary non-zero vector which we split (uniquely) as
z = Nz˜ + z′ where NTz′ = 0. If z′ = 0, we have zTMz = z˜TNTM1Nz˜ > 0
since CTN has full rank. If z′ 6= 0, we have zTMz = zTM1z + z′TM2z′ > 0
since zTM1z ≥ 0 and z′TM2z′ > 0. An analogous argument holds for the
matrix T .
We will now outline a few specific and valid choices that are implemented in
the basic flow solvers of MRST. We start by considering the one-dimensional case
with the cell [−1, 1] for which N = C = [1,−1]T and Q⊥N = Q⊥C = 1√2 [1, 1]
T.
Hence
M =
1
2
[
1
−1
]
1
K
[1,−1] + 1
2
[
1
1
]
SM [1, 1],
T =
1
2
[
1
−1
]
K [1,−1] + 1
2
[
1
1
]
S [1, 1].
(17)
15
The structure of the inner product should be invariant under scaling of K and
thus we can write the inner product as a one-parameter family of the form
M =
1
2K
([
1 −1
−1 1
]
+
2
t
[
1 1
1 1
])
,
T =
K
2
([
1 −1
−1 1
]
+
t
2
[
1 1
1 1
])
.
(18)
In the following, we will use this one-dimensional expression together with trans-
formation properties of the inner product to look at the correspondence between
mimetic, TPFA, and Raviart–Thomas methods.
Two-point type methods. The TPFA method is the gold standard for prac-
tical reservoir simulation despite its theoretical shortcomings. Since the TPFA
discretisation requires a diagonal tensor, it is easily seen from (11) that this is
only possible in the case when the vectors K~ni and ~ci are parallel, i.e., the grid
is K-orthogonal. In any case (K-orthogonality or not), we define the diagonal
TPFA transmissibility tensor T by
T ii = ~ni ·K~ci/|~ci|2. (19)
This defines the unique TPFA method for K-orthogonal grids3. The extension
to non-orthogonal grids is not unique, and the TPFA method does not give
a consistent discretisation in this case.Because of its simplicity, the TPFA
method is strictly monotone if Tii > 0, which implies that the fluxes form a
directed acyclic graph, a property that can be used to accelerate the solution of
the transport equations considerably, as discussed in [28].
When written in its standard form, the TPFA method is cell-centred and
thus less computationally costly than face-centred methods that arise from a
consistent hybrid mimetic formulation. However, since the method is not con-
sistent, it will in many cases be necessary to investigate the grid-orientation
effects of the TPFA method before using it for realistic simulations. We there-
fore present a mimetic-type discretisation that coincides with the TPFA method
in its region of validity, while at the same time giving a valid mimetic inner prod-
uct for all types of grids and permeability tensors. This minimises the need for
investigating other effects of the TPFA method, such as errors introduced by
corners and well modelling. An advantage of this method compared to using,
e.g., an MPFA method is that the implementation is simpler for general unstruc-
tured grids. We refer to the corresponding method as IP QTPFA. To derive the
method, we consider a cuboid cell and insert (10) into (9) defined for a single
face to obtain
Tii~ci · ~a = ~niK~a.
3As written, this does not always yield a positive value for the two-point transmissibility.
To ensure positive transmissibilities it is normal to define the face centroids as the arithmetic
mean of the associated corner-point nodes and the cell centroids as the arithmetic mean of the
top and bottom surface centroids. For most reservoir grids this gives a positive transmissibility.
16
c1 c2
c3
c4
n1 n2
n3
n4
~c1 + ~c2 = 0, ~c3 + ~c4 = 0
=⇒ Q⊥C =
1√
2

1 0
1 0
0 1
0 1

~n1 + ~n2 = 0, ~n3 + ~n4 = 0
Figure 6: Creation of nullspace of C for a parallelepiped. The left plot shows the
vectors ~ci and ~ni (the latter scaled by a factor 0.3).
Next, we set ~a = ~ni and use the property that ~ci · ~n = 12 |Ωi| for cuboid cells to
derive an expression for Tii, which can be written in matrix form and equated
with (14)
TTPF =
2
|Ωi| diag(NKN
T) =
1
|Ωi|NKN
T + T 2. (20)
Here, T 2 should be invariant under P⊥C . Moreover, one can easily show that
P⊥CN = 0 for parallelepipeds (see Figure 6), which means that we can define
the generalised TPFA inner product as
T =
1
|Ωi|
[
NKNT + 2P⊥C diag(NKN
T)P⊥C
]
,
M = T−1.
(21)
Equation (1) is invariant when space and permeability are transformed as
~x 7→ S~x and K 7→ STKS, (22)
respectively. We emphasise that the inner product (21) evidently is invariant
under the transformations (22), but will only be diagonal for K-orthogonal grids.
In the one-dimensional case, (21) simplifies to
T =
[
K 0
0 K
]
, (23)
which is the TPFA expression for this case. The same result is obtained by
setting t = 2 in the one-dimensional expression (18). Since the mimetic and
the TPFA method do not couple dimensions for orthogonal grids with diagonal
tensor, this derivation is also valid for rectangular cuboids.
Raviart–Thomas (RT0). The following inner product is equivalent to the
mixed Raviart–Thomas inner product for grids that are orthogonal and has the
same principal axes as the permeability tensor
M =
1
|Ωi|CKC
T +
|Ωi|
6
P⊥N
[
diag(NKNT)
]−1
P⊥N . (24)
17
This can be verified by a direct calculation for the one-dimensional case, which
reduces to
M =
1
K
[
2/3 −1/3
−1/3 2/3
]
, (25)
which coincides with (18) for t = 6. The fact that this inner product does not
couple different directions for orthogonal grids and diagonal tensors implies that
(24) is also equal to the lowest-order Raviart–Thomas (RT0) method on (rectan-
gular) cuboids. We refer to this inner product as ‘IP QRT’. The corresponding
quasi-inverse, which is the exact inverse for orthogonal grids, reads
T =
1
|Ωi|
[
NKNT + 6P⊥C diag(NKN
T)P⊥C
]
. (26)
This inner product will also, by the transformation property, be equal to the
Raviart–Thomas formulation for all cases that can be transformed to the above
case by an affine transformation of the form (22). Using a mimetic inner product
which is simpler to calculate and equal to the mixed RT0 inner product for all
grid cells is not possible because of the need to integrate the nonlinear function
introduced by the determinant of the Jacobian of the mapping from the grid
cell to the unit cell where the RT0 basis functions are defined.
Local-flux mimetic MPFA. In addition to the above methods, which are
all based on the assumption of a positive-definite inner product and exactness
for linear flow, the MPFA method can be formulated as a mimetic method.
This was first done by [21, 24] and is called the local-flux mimetic formulation
of the MPFA method. In this case, all faces of a cell are such that each corner
is associated with d unique faces, where d is the dimension. The inner product
of the local-flux mimetic method gives exact result for linear flow and is block
diagonal with respect to the faces corresponding to each corner of the cell, but it
is not symmetric. The block-diagonal property makes it possible to reduce the
system into a cell-centred discretisation for the cell pressures. This naturally
leads to a method for calculating the MPFA transmissibilities. The crucial point
is to have the corner geometry in the grid structure and handle the problems
with corners which do not have three unique half faces associated. Currently
the MPFA-O method is formulated in such a way [21] and work has been done
for the MPFA-L method, but in this case the inner product will vary over the
different corners. Our implementation of MPFA is based upon a combination
of Matlab and C and is therefore available as an add-on module to MRST.
Parametric family. We notice that the (inverse) inner products IP QTPFA
in (21) and IP QRT in (26) differ only by a constant in front of the regularisation
part (the second term). Both methods belong to a family whose inverse inner
product can be written in the form
T =
1
|Ωi|
[
NKNT + tP⊥C diag(NKN
T)P⊥C
]
M = T−1,
(27)
18
in which t is a parameter that can be varied continuously from zero to infinity. In
MRST, this family of inner products is called ‘IP QFAMILY’ and the parameter
t is supplied in a separate option
S = computeMimeticIP(G, rock, 'Verbose', true,...
'InnerProduct', ' ip qfamily ' , 'qparam', t);
IP SIMPLE. For historical reasons, the default inner product used in MRST
reads
Q = orth(A−1N)
M =
1
|Ωi|CK
−1CT +
d|Ωi|
6 tr(K)
A−1(I −QQT)A−1 (28)
with the approximate inverse
Q = orth(AC)
T =
1
|Ωi|
[
NKNT +
6
d
tr(K)A(I −QQT)A
]
.
(29)
This inner product was used in [4] inspired by [9] and was chosen to resemble the
mixed Raviart–Thomas inner product (they are equal for scalar permeability on
orthogonal grids, which can be verified by inspection. Since this inner product
is based on velocities, it involves pre- and post-multiplication of (inverse) face
areas, and it might not be obvious that it fits into the formulations (15)–(16).
However, after a small computation one is convinced that the second part of
(15) is invariant under multiplication by P⊥N so its eigenspace corresponding
to the nonzero eigenvalues must be equal to the nullspace of NT. A similar
argument holds for the inverse inner product.
Example 2 (Grid-Orientation Effects and Monotonicity) It is well known
that two-point approximations are convergent only in the special case of K-
orthogonal grids. Mimetic and multipoint schemes, on the other hand, are
constructed to be consistent and convergent for rough grids and full-tensor per-
meabilities, but may suffer from pressure oscillations for full-tensor permeabil-
ities and large anisotropy ratios and/or highaspect ratio grids. In this exam-
ple, we use one of the example grids supplied with MRST to illustrate these
two observations. The routine twister normalises all coordinates in a rec-
tilinear grid to the interval [0, 1], then perturbs all interior points by adding
0.03 sin(pix) sin(3pi(y−1/2)) to the x-coordinates and subtracting the same value
from the y-coordinate before transforming back to the original domain. This cre-
ates a non-orthogonal, but smoothly varying, logically Cartesian grid.
To investigate grid-orientation effects, we impose a permeability tensor with
anisotropy ratio 1 : 1000 aligned with the x-axis and compute the flow resulting
from a prescribed horizontal pressure drop and no-flow conditions on the top and
bottom boundaries. Figure 7 shows pressure computed on a 100 × 100 grid by
19
0 50 100
0
20
40
60
80
100
TPFA
0 50 100
0
20
40
60
80
100
mimetic
0 50 100
0
20
40
60
80
100
MPFA
0 50 100
0
20
40
60
80
100
x
0 50 100
0
20
40
60
80
100
x
0 50 100
0
20
40
60
80
100
x
 
 
20
30
40
50
60
70
80
90
Figure 7: Grid-orientation effects for a homogeneous domain with Dirichlet boundary
conditions (left,right) and no-flow conditions (top, bottom) computed with three differ-
ent pressure solvers in MRST. The permeability field is homogeneous with anisotropy
ratio 1 : 1000 aligned with the grid. The upper row shows streamlines visualising the
velocity field, whereas the bottom row shows the pressure field.
the TPFA method, the mimetic method with inner product IP SIMPLE, and the
local-flux mimetic version of the MPFA-O method. To visualise the correspond-
ing velocity field, we also show streamlines traced by MRST’s implementation
of Pollock’s method. Whereas the mimetic and MPFA-O schemes produce the
expected result, the pressure and velocity solutions computed by TPFA show sig-
nificant grid-orientation effects and are obviously wrong.
In Figure 8, we have repeated the experiment with the tensor rotated by
pi/6 on a grid with 11 × 11 cells. Again, we see that the solution computed
by the TPFA scheme is wrong. In this case, the pressures computed with the
mimetic and the MPFA-O schemes appear to be correct and have no discernible
oscillations. However, the streamline distributions clearly show that the resulting
velocity fields are highly oscillatory, in particular for the mimetic method with
the IP SIMPLE inner product.
To compare the effect of using an approximate inverse in the inner-product
matrices, we have computed the velocity field for the mimetic methods resulting
from inner products (29), (24), and (21). From the streamline plots shown in
Figure 9, it is evident that both IP SIMPLE and IP QRT yield more oscillatory
velocity fields than the IP QTPF method.
Example 3 (Analytical solution) In this example, we use a classical analyt-
ical solution based on complex analysis to verify the numerical solutions obtained
20
0 5 10
0
2
4
6
8
10
TPFA
0 5 10
0
2
4
6
8
10
mimetic
0 5 10
0
2
4
6
8
10
MPFA
0 5 10
0
2
4
6
8
10
x
0 5 10
0
2
4
6
8
10
x
0 5 10
0
2
4
6
8
10
x
 
 
20
40
60
80
100
Figure 8: Monotonicity effects demonstrated with the same setup as in Figure 7, but
with the anisotropy ratio 1 : 1000 making an angle pi/6 with the grid directions.
SIMPLE QRT QTPFA
Figure 9: Lack of monotonicity visualised by tracing streamlines for the velocity fields
computed by three mimetic pressure solvers in MRST for the same setup as in Figure 8.
21
by MRST. A standard method to construct analytical solutions to the Laplacian
equation in 2D is to write the unknown as the real or imaginary part of an com-
plex analytic function f(z). We choose the function f(z) = (z + 12z
2) and set
our analytical solution to be pa(x, y) = If(x + iy) = y + xy. As our computa-
tional domain, we choose a 2D triangular grid generated from the seamount.mat
data set (see e.g., the upper-left plot in Figure 2) and its dual Voronoi grid, both
centred at the origin. By prescribing pa(x, y) along the boundary, we obtain a
Dirichlet problem with known solution.
To compute numerical approximations to pa(x, y) we use the MPFA-O scheme
and the mimetic method with inner product ’IP SIMPLE’. Figure 10 shows the
approximate solutions and the discrepancy from the analytical solution at the
cell centres for both grids. For the triangular grid, the two methods produce the
exact same result because all mimetic methods with a symmetric inner product
are identical for cell and face pressures on triangular grids. On the Voronoi
grid, the mimetic method is slightly more accurate than the MPFA-O method
and both methods have lower errors than on the triangular grid.
Example 4 (Real-Field Model) In this example, we consider two synthetic
models derived from the 3D simulation model of a reservoir from offshore Nor-
way. The original model is given as a 46× 112× 22 logically Cartesian corner-
point grid with 44,915 active cells. To assess grid-orientation effects introduced
by different inner products, we consider two single layers (layer one and ten
from the top of the model), which we flatten and modify so that the thickness is
constant and all pillars in the corner-point description are vertical. Well posi-
tions are assigned by keeping one perforation for each of the original wells. Let
(xi, yi), i = 1, . . . , n denote the resulting well positions and qi the well rates. If
we model each well as a logarithmic singularity along a vertical line at (xi, yi),
the corresponding analytical solution to (1) in an infinite domain with constant
permeability K is given by
p(x, y, z) =
n∑
i=1
qi
2piK
ln
(√
(x− xi)2 + (y − yi)2
)
. (30)
To generate a representative analytical solution, we set K = 500 mD and
prescribe (30) along the outer lateral boundary of each model. In Figure 11,
we compare this analytical solution with numerical solution computed by the
TPFA method and the mimetic IP QRT method for the first layer. Whereas
the mimetic solution shows good correspondence with the analytical solution,
the TPFA solution exhibits large grid-orientation effects, which are particularly
evident between wells one and four and in the region below well two. With the
consistent mimetic scheme, the error is localised around the wells, whereas it is
up to 10 % and distributed all over the reservoir for the TPFA method. (The
full 3D model has a much rougher geometry that contains pinch-outs, sloping
faults, and eroded layers, which all will contributed to larger grid-orientation
effects for inconsistent methods). In a real simulation, a good well model e.g.,
as discussed in the next section) can correct for errors in the vicinity of the well,
but not global errors as in the inconsistent TPFA method.
22
 
 
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
 
 
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
pmim(x, y) on PEBI grid pmim(x, y) on triangular grid
 
 
−4
−3
−2
−1
0
1
2
3
x 10−4
 
 
−6
−4
−2
0
2
4
6
8
x 10−4
pmim(x, y)− pa(x, y) pmim(x, y)− pa(x, y)
 
 
−2
−1
0
1
2
x 10−4
 
 
−6
−4
−2
0
2
4
6
8
x 10−4
pmpfa(x, y)− pa(x, y) pmpfa(x, y)− pa(x, y)
Figure 10: The upper row shows the pressure for a Dirichlet problem with analytical
solution pa(x, y) = y + xy computed by a mimetic method on a PEBI grid and on a
triangular grid. The second and third rows show the error in solutions computed by
the mimetic and MPFA-O method, respectively.
23
3
5
1
4
Analytic
2
3
5
1
4
TPFA
23
5
1
4
Mimetic
2
 
100
200
300
4.56 4.58 4.6 4.62
x 105
7.32
7.321
7.322
7.323
7.324
7.325
7.326
7.327
x 106
 
3
TPFA
5
1
4
2
 
−10
−5
0
5
4.56 4.58 4.6 4.62
x 105
7.32
7.321
7.322
7.323
7.324
7.325
7.326
7.327
x 106
 
3
Mimetic
5
1
4
2
 
−1
−0.5
0
0.5
1
1.5
4.56 4.57 4.58 4.59 4.6 4.61
x 105
7.32
7.321
7.322
7.323
7.324
7.325
7.326
7.327
x 106
 
4
3
7
TPFA
1
5
8
62
 
−10
−5
0
5
10
4.56 4.57 4.58 4.59 4.6 4.61
x 105
7.32
7.321
7.322
7.323
7.324
7.325
7.326
7.327
x 106
 
4
3
7
Mimetic
1
5
8
62
 
−2
−1
0
1
2
3
Figure 11: Analytical and numerical solutions for the flattened top layer of a real-
field model. The upper row compares pressure solutions (plotted in units ’bar’) for
layer one, whereas the two lower rows show the cell-wise discrepancies between the
analytical and the numerical solutions for layers one and ten.
24
Table 1: Error as a function of inner product in (27) for the real-field models. The
error is defined as the maximum discrepancy in cell pressure scaled by the total span
in analytic pressure values outside the well cells. The corresponding errors for the
IP QRT, MPFA-O, and TPFA methods are 0.0096, 0.0158, and 0.0560 for layer one
and 0.0127, 0.0123 and 0.053 for layer ten.
Inner product (t) 0.5 1 2 3 4
Layer one 0.0752 0.0394 0.0161 0.0097 0.0080
Layer ten 0.0695 0.0326 0.0119 0.0095 0.0104
Inner product (t) 5 6 7 10 100
Layer one 0.0078 0.0096 0.0118 0.0112 0.0179
Layer ten 0.0114 0.0127 0.0137 0.0152 0.0189
In Table 1 we report the maximum relative error in pressure for various
members of the inner-product family introduced above applied to the grid ex-
tracted from the tenth layer. For layer ten, the minimum error occurs around
t = 3, which is a method somewhere between the quasi-two-point and the quasi-
Raviart–Thomas methods. For both grids, the minimal error is smaller than for
both the IP QRT and the MPFA-O method.
Using the full model, we have also verified (or validated) our implementations
against the leading commercial simulator for incompressible, two-phase flow.
3.2 Finite-Difference Stencils
To complete the discussion of the previously presented mimetic finite-difference
methods we demonstrate a few of the resulting finite-difference stencils that re-
late the interface pressure values in (4). In particular, we discretise the pressure
equation (1) with permeability K = I on a two-dimensional, equidistant grid,
∆x = ∆y = 1 using inner products already implemented in MRST. Figure 12
shows the results. We notice in particular that the ‘IP SIMPLE’ inner product
of equation (29) and the ‘IP QFAMILY’ inner product of equation (27) with pa-
rameter t = 6 produce stencils that have both positive and negative off-diagonal
coupling terms. Moreover, setting t = 4 produces a stencil that resembles the
classical five-point discretisation scheme of the Laplace operator on a rotated
grid. Finally, for t ∈ (2, 4), all off-diagonal coupling terms are negative. The
case of t = 2 is identical to the inner product given by ‘IP TPFA’ and ‘IP
QTPFA’. For general grids and permeability, the inner product given by (27)
only includes ‘IP QRT’ and ‘IP QTPFA’ since these inner products are the
only ones that are invariant under the affine transformations in (22).
3.3 Well Modelling
Wells in reservoirs typically have small diameters compared to the size of the
simulation cells, and it is therefore common to employ a well index (productivity
index) WI to relate the local flow rate q to the difference between the well
25
1/2 5 1/2
−3/2 −3/2
−3/2 −3/2
(a) IP SIMPLE, IP QFAMILY (t = 6)
0 4 0
−1 −1
−1 −1
(b) IP QFAMILY (t = 4)
−1/4 7/2 −1/4
−3/4 −3/4
−3/4 −3/4
(c) IP QFAMILY (t = 3)
−1/2 3 −1/2
−1/2 −1/2
−1/2 −1/2
(d) IP TPF, IP QFAMILY (t = 2)
Figure 12: Finite-difference stencils for interface pressures when ∆x = ∆y = 1
and K = I.
pressure pw and numerically computed pressure pE in the perforated grid-cell
as follows
−q = λt(sE)WI(pE − pw). (31)
Commonly used is Peaceman’s well index [30], which for a vertical well in a
Cartesian cell with dimensions ∆x×∆y ×∆z is given as
WI =
2pik∆z
ln(r0/rw)
. (32)
For isotropic media, k is given by K = kI, and
r0 = 0.14(∆x2 + ∆y2)
1
2 , (33)
where r0 is the effective well-cell radius and can be interpreted as the radius
at which the actual pressure equals the numerically computed pressure. The
validity of the Peaceman well-index decreases rapidly with increasing near-well
heterogeneity and grid skewness. It is also important to note that the Peaceman
well-index is developed for the TPFA method and is not valid for other methods
(such as MFE method with exact integration or mimetic methods in general)
because different numerical methods in general give different well-cell pressures.
We will now give a short description of how to extend Peaceman’s results to
other methods than TPFA; for a more extensive study of well models, we refer
to [23].
Assuming steady-state, single-phase, horizontal flow in a homogeneous and
isotropic reservoir with radial flow near the well, there exists an analytical flow
26
Table 2: The constant C(IP, β) in the formula (37) for r0 for different mimetic
inner products as a function of the aspect ratio β of the well cell.
β 1 2 4 8 16 32 64
IP QFAMILY(t=1) 0.0495 0.0546 0.0655 0.0756 0.0823 0.0862 0.0883
IP TPF 0.1404 0.1404 0.1404 0.1404 0.1404 0.1404 0.1404
IP QFAMILY(t=3) 0.2015 0.1959 0.1857 0.1777 0.1729 0.1703 0.1690
IP QFAMILY(t=4) 0.2423 0.2328 0.2153 0.2019 0.1939 0.1896 0.1874
IP QFAMILY(t=5) 0.2712 0.2588 0.2363 0.2189 0.2087 0.2032 0.2003
IP SIMPLE 0.2926 0.2782 0.2518 0.2316 0.2197 0.2133 0.2100
model near the well given by
p(r) =
qµ
2pik∆z
ln
( r
rw
)
+ pw. (34)
We refer to [27, pages 150–153] for a derivation of this expression. Peaceman
used the five-spot pattern problem to derive both numerical and analytical ex-
pressions for r0 based on the relation above. However, the equivalent radius can
also be calculated by simulating an infinite reservoir with one well and radial
flow, and we employ this method to calculate new equivalent radii for mimetic
methods.
The radial flow simulation is done by forcing appropriate flux boundary
conditions. Consider a section of the reservoir containing a well as displayed
in Figure 13 where we prescribe Fi = qθi/2pi as flux boundary conditions on a
face i. When the resulting system is solved for pressure and flow, we use the
analytical pressure relation (34) with rw = r0 so that pw = pE , and set r = rb
to be the distance from the well cell to the centroid of a boundary cell,
p(rb) =
qµ
2pik∆z
ln
( rb
r0
)
+ pE . (35)
We then substitute p(rb) by pb, the numerically computed pressure in the bound-
ary cell (at rb), and solve (35) for r0,
r0 = rb exp
( qµ
2pik∆z(pE − pb)
)
. (36)
If we assume that the equivalent radius for mimetic methods is on the same form
as for TPFA shown in (33), then r0 will depend on the actual inner product and
on the grid aspect ratio of the well cell, β = ∆x/∆y, i.e.,
r0(IP, β) = C(IP, β)(∆x2 + ∆y2)
1
2 . (37)
Thus, if we know r0 we can easily compute C(IP, β), which is given in Table 2
as a function of β for the IP TPF and IP SIMPLE inner products and selected
members of the IP QFAMILY(t) inner-product family (27).
27
Figure 13: Section of infinite reservoir with well and radial flow.
4 Multiscale Pressure Solvers
A main purpose of MRST is to provide an efficient toolkit for the development
and testing of new ideas. In this section, we present examples in which the
solvers describedin the previous sections are applied as a building block to
develop multiscale pressure solvers. Given the importance of grid geometry
on the quality of numerical computations, it is crucial to have flexible tools
that allow testing new algorithms on many different types of polyhedral grids
of varying complexity in 3D. Through the discussion of multiscale solvers, we
hope to give the reader a taste of the utility of the toolbox.
Multiscale flow solvers [17, 15] can be seen as numerical methods that take
a fine-scale model as input, but solve for a reduced set of unknowns defined
on a coarse grid to produce a solution that has both coarse-scale and fine-scale
resolution. A key characteristic of multiscale methods is the incorporation of
fine-scale information into a set of coarse-scale equations in a way that is consis-
tent with the local property of the differential operator. Generally, a multiscale
method uses two operators: a compression (or restriction) operator that takes
information from the fine scale to the coarse scale, and a reconstruction (or
prolongation) operator that takes information from the coarse scale to the fine
scale. In particular, the compression operator reduces the system of discretised
flow equations on a fine grid to a system with significantly fewer unknowns
defined on a coarse grid. Similarly, the reconstruction operator constructs an
approximate fine-scale solution from the solution computed on the coarse scale.
The reconstruction operators are typically computed numerically by solving a
localised flow problem as in an upscaling method.
Different multiscale flow solvers are distinguished by how they define their
degrees of freedom and the compression and reconstruction operators. In the
multiscale finite-volume (MsFV) method [18, 32], the coarse-scale degrees-of-
freedom are associated with pressure values at the vertices of the coarse grid.
The reconstruction operator is associated with pressure values and is defined by
solving flow problems on a dual coarse grid. (In addition, the method needs to
solve a local flow problem on the primal coarse grid to recover conservative fine-
scale fluxes). In the multiscale mixed finite-element (MsMFE) method [11, 4],
28
the coarse-scale degrees-of-freedom are associated with faces in the coarse grid
(coarse-grid fluxes) and the reconstruction operator is associated with the fluxes
and is defined by solving flow problems on a primal coarse grid. In the following,
we will present the MsMFE method in more detail. To this end, we will use a
finite-element formulation, but the resulting discrete method will have all the
characteristics of a (mass-conservative) finite-volume method.
The multiscale method implemented in MRST is based on a hierarchical
two-level grid in which the blocks Ωi in the coarse simulation grid consist of a
connected set of cells from the underlying fine grid, on which the full heterogene-
ity is represented. In its simplest form, the approximation space consists of a
constant approximation of the pressure inside each coarse block and a set of ve-
locity basis functions associated with each interface between two coarse blocks.
Consider two neighbouring blocks Ωi and Ωj , and let Ωij be a neighbourhood
containing Ωi and Ωj . The basis functions ~ψij are constructed by solving
~ψij = −K∇pij , ∇ · ~ψij =

wi(x), if x ∈ Ωi,
−wj(x), if x ∈ Ωj ,
0, otherwise,
(38)
in Ωij with ~ψij ·~n = 0 on ∂Ωij . If Ωij 6= Ωi∪Ωj , we say that the basis function is
computed using overlap or oversampling. The purpose of the weighting function
wi(x) is to distribute the divergence of the velocity, ∇ · ~v, over the block and
produce a flow with unit flux over the interface ∂Ωi ∩ ∂Ωj , and the function
is therefore normalised such that its integral over Ωi equals one. Alternatively,
the method can be formulated on a single grid block Ωi by specifying a flux
distribution (with unit average) on one face and no-flow condition on the other
faces, see [2] for more details. In either case, the multiscale basis functions—
represented as vectors Ψij of fluxes—are then collected as columns in a matrix
Ψ, which will be our reconstruction operator for fine-scale fluxes. To define
the compression operator, we introduce two prolongation operators I and J
from blocks to cells and from coarse interfaces to fine faces, respectively. The
operator I is defined such that element ij equals one if block number j contains
cell number i and zero otherwise; J is defined analogously. The transposed of
these operators will thus correspond to the sum over all fine cells of a coarse block
and all fine-cell faces that are part of the faces of the coarse blocks. Applying
these compression operators and ΨT to the fine-scale system, we obtain the
following global coarse-scale system ΨTBΨ ΨTCI ΨTDJITCTΨ 0 0
J TDTΨ 0 0
 uc−pc
pic
 =
 0ITq
0
 . (39)
Once (39) is solved, the fine-scale fluxes can be obtained immediately as v =
Ψuc. The basic steps of the multiscale algorithm are summarised in Figure 14.
More details about how to include wells in a appropriate way is given in [31].
Having introduced the multiscale method, we should explain how to use it.
To this end, we consider a simple example.
29
Figure 14: Key steps of the multiscale method: (1) blocks in the coarse grid are
defined as connected collections of cells from the fine grid; (2) a local flow problem is
defined for all pairs of blocks sharing a common face; (3) the local flow problems are
solved and the solutions are collected as basis functions (reconstruction operators); and
(4) the global coarse-system (39) is assembled and solved, then a fine-scale solution
can be reconstructed.
Example 5 (Log-Normal Layered Permeability) In this example, we will
revisit the setup from the previous section. However, we neglect gravity and
instead of assuming a homogeneous permeability, we increase the number of
cells in the vertical direction to 20, impose a log-normal, layered permeability
as shown in Figure 3, and use the layering from the permeability to determine
the extent of the coarse blocks; a large number of numerical experiments have
shown that the MsMFE method gives best resolution when the coarse blocks
follow geological layers [4]. In MRST, this amounts to:
[K, L] = logNormLayers([nx, ny, nz], [200 45 350 25 150 300], 'sigma', 1);
:
p = processPartition(G, partitionLayers(G, [Nx, Ny], L) );
:
plotCellData(G,mod(p,2));
outlineCoarseGrid(G,p,'LineWidth',3);
The permeability and the coarse grid are shown in Figure 15. Having par-
titioned the grid, the next step is to build the grid structure for the coarse grid
and generate and solve the coarse-grid system.
CG = generateCoarseGrid(G, p);
CS = generateCoarseSystem(G, rock, S, CG, ...
ones([G.cells.num, 1]),'bc' , bc, ' src ' , src);
xMs = solveIncompFlowMS(initResSol(G, 0.0), G, CG, ...
p, S, CS, fluid,'src ' , src, 'bc' , bc);
The multiscale pressure solution is compared to the corresponding fine-scale
solution in Figure 15. The solutions appear to be quite close in the visual norm.
For single-phase problems, a multiscale flow solver without any form of par-
allelism will have a computational cost that is comparable to that of a fine-scale
flow solver equipped with an efficient linear routine, i.e., Matlab’s built-in
30
 
Permeability
 
 50
100
150
200
250
300
350
400
450
Coarse partition
0
10
20
0
10
20
0
10
20
x
Fine−scale pressure
y 0
10
20
0
10
20
0
10
20
 
x
Multiscale pressure
y 
10.1
10.2
10.3
10.4
10.5
Figure 15: Comparison of the MsMFE and fine-scale mimetic flow solvers for the setup
from Figure 4. The top plots show the layered permeability field and the corresponding
partitioning into coarse blocks. The lower plots show the fine-scale pressure solution(left) and the multiscale solution (right).
solvers for small problems and e.g., the AGMG solver [29] for large problems.
The advantage of a multiscale solver comes first when considering multiphase
flow problems, where the key to computational efficiency is reuse of previous
computations. For a multiphase system, the basis functions will become time-
dependent when K is replaced by λK in (38), but this time-dependence is weak
for an incompressible system, and the basis functions can therefore be computed
initially and updated infrequently throughout a simulation. Solving the coarse-
scale problem (39) is usually inexpensive compared to solving the corresponding
fine-scale system or computing basis functions. In the current release of MRST,
gravity is only resolved on the coarse scale. However, gravity effects can easily
be included also on the fine scale by using special correction functions.
Next, we present a short example that demonstrates the applicability of the
MsMFE method on unstructured grids in 3D. Although several of our papers
have argued that the method can easily be formulated on fully unstructured
grids in 3D, the following is the first example demonstrating the actual capability
on grids that do not have an underlying logically Cartesian structure.
Example 6 (Prismatic and 2.5D PEBI Grids) The MsMFE method was
previously demonstrated on Cartesian and logically Cartesian (corner-point)
grids, see [3, 20, 4]. However, the method is equally feasible on general, un-
structured grids provided there exists a suitable partitioning of the underlying
(fine-scale) grid. Figure 16 shows the solution of the single-phase flow problem
(1) with isotropic, homogeneous K on a computational domain Ω that has been
31
Triangular coarse grid Multiscale pressure solution
PEBI coarse grid Multiscale pressure solution
Figure 16: Multiscale discretisation of a flow problem with linear pressure drop
on a prismatic and a PEBI grid.
discretised using both prismatic and PEBI cells. These cells are subsequently
partitioned into a coarse grid by means of the well-known software package
METIS [19]. For simplicity, the areal PEBI grid was created as the Voronoi
diagram of the triangular grid used to extrude the prismatic grid and hence the
fault is more irregular on the PEBI grid. Despite the irregular block boundaries,
the multiscale method is able to accurately capture the linear pressure drop from
the west to the east boundaries on both grids.
The public version of MRST currently supports single-phase and two-phase
flow. Our in-house version of MRST has compressible black-oil fluid models im-
plemented, along with several flow and transport solvers. In the last example,
we will demonstrate the use of one of our experimental multiscale flow solvers to
simulate compressible flow on a geological model with industry-standard com-
plexity.
Example 7 (Primary Production from a Gas Reservoir) In this exam-
ple, we will briefly discuss the use of the MsMFE method to compute primary
production from a gas reservoir. As our geological model, we will use one of
32
Permeability
Coarse grid
Rate in well perforation (m3/day)
100 200 400 600 800 1000
535
540
545
550
555
560
 
 
Reference
5⋅10−2
5⋅10−4
5⋅10−6
5⋅10−7
Figure 17: Primary production from a gas reservoir computed by the MsMFE method.
the shallow-marine realisations from the SAIGUP study [25]. One approach for
simulating compressible black-oil models with the MsMFE method, is to use a
mixed residual formulation[
B C
CT P
] [
ums + uˆν+1
pms + pˆ
ν+1
]
=
[
0
Ppn + V uν
]
.
Here, elliptic multiscale basis functions computed from (38) using a weight func-
tion wi(x) that scales like the porosity act as predictors. Compressibility effects
are accounted for by a parabolic correction that is computed using a standard
(non)overlapping Schwarz domain-decomposition method. The method then it-
erates on the multiscale solution and the corrections until the fine-scale residual
is below a prescribed tolerance. The geological model consists of 40 × 120 × 20
cells (see Figure 17) and is assumed to be initially filled with an ideal gas at
200 bar pressure. The reservoir is produced by a single producer, operating at a
constant bottom-hole pressure of 150 bar. Figure 17 compares the fine-scale ref-
erence solution with multiscale solutions for different tolerances; as our measure,
we have used the rate in the well perforation. Except for the coarsest tolerance,
5 · 10−2, the multiscale solution appears to overlap with the fine-scale reference
solution.
Many reservoir management challenges can be cast as mathematical optimi-
sation problems. Examples include data assimilation where a misfit function is
to be minimised, and finding optimal well controls to maximise some economic
performance measure. A popular approach to obtaining objective function gra-
dients is the use of adjoint equations. The adjoint system of equations is the
transpose of the linearised forward system, and accordingly has similar com-
plexity as the forward simulation. Our in-house version of MRST includes an
adjoint code for a sequential forward simulation using the mimetic discretisation
33
for the pressure equation and an implicit version of the scheme (7) for satura-
tion. In addition, an adjoint code for the multiscale method combined with the
flow based coarsening approach is included, see [22].
Example 8 (Optimising Net Present Value (NPV)) In this example we
consider a synthetic model with two horizontal injectors and one multilateral
producer, see Figure 18. We attempt to maximise a simple NPV function; the
oil revenue is $100 per barrel and our costs are realised through water injection
and production, each $10 per barrel. This means that when the water cut in the
producer exceeds ≈ 0.82, we are no longer making money. We compare three
production strategies:
1. The producer operates at constant BHP until the water cut reaches 0.82,
and the well is shut.
2. We assume that each producer well-segment is equipped with an inflow
control device (ICD). The producer operates at constant BHP and when-
ever the water cut in a segment exceeds 0.82, the corresponding ICD shuts.
The process is continued until all ICDs are shut.
3. We use the adjoint code to find optimal segment rates corresponding to a
local maximum of the NPV-function.
The initial simulation input is constant and equal rate for the two injectors and
constant BHP for the producer. The initial simulation time is set to 500 days
which is equivalent to 1.25 PVI. In Figure 18 we show the accumulated NPV for
the three production scenarios. We observe that for production scenario 1, the
well is shut down after about 225 days with an achieved NPV of $ 72.8 million.
Scenario 2 is equal to scenario 1 until the first ICD is shut, and the improvement
obtained by being able to shut down individual segments of the well is evident.
All ICDs are shut after about 375 days, and the obtained NPV is $ 79.9 million.
Finally, in scenario 3, water cut is not taken into account, only maximising the
NPV. The obtained NPV is $ 92.6 million.
5 Outlook
Replicablility is a fundamental scientific principle that is currently receiving
increasing attention among computational scientists. The level of sophistication
and complexity in new computational methods makes it difficult for researchers
to implement methods published by others and replicate their results.
We believe that releasing theMatlab Reservoir Simulation Toolbox (MRST)
under the GNU General Public License (GPL) is an important step for our group
towards supporting the idea of reproducible science, thereby allowing others to
more easily benefit from our efforts. In this paper we have given an overview
of MRST and discussed in detail a family of consistent methods that arecon-
vergent on fully unstructured, polyhedral grids. We have also demonstrated the
compressible black-oil and adjoint multiscale features that currently exists only
34
Permeability
Initial saturation and wells
Accumulated NPV
0 100 200 300 400 500
0
2
4
6
8
10 x 10
7
Time [days]
N
PV
 [$
]
 
 
Reactive
Reactive − ICD
Optimal − ICD
Figure 18: Accumulated net present value for the three production strategies.
in our in-house, development version of the package. Although we are not yet
there, we are aiming towards a state where our publications on new computa-
tional methodologies can be accompanied by open-access software that can be
used by others to verify our computational results.
6 Acknowledgements
The research was funded in part by the Research Council of Norway through
grants no. 158908, 174551, 175962, 178013, and 186935.
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